in Artificial Life VIII, Standish, Abbass, Bedau (eds)(MIT Press) 2002. pp 186–191 1

The Origins of Mimicry Rings

Daniel W. Franks and Jason Noble School of Computing, University of Leeds Email: [dwfranks|jasonn]@comp.leeds.ac.uk

Abstract Although beneficial to the mimic, Batesian mimicry is Mutualistic Mullerian¨ mimicry and parasitic Batesian detrimental to the survival of the model, as the presence mimicry can co-exist in mimicry rings, i.e., mimetic re- of tasty individuals dilutes the honesty of the warning lationships between multiple species. Theory suggests coloration. This means that Batesian mimicry is a par- that all Mullerian¨ mimics in an ecosystem should con- asitic relationship between mimic and model (see e.g., verge into one large ring. Potentially, the presence of Wickler, 1968) . Batesian mimics will encourage convergence. It has been suggested that rare species should seek out common Predators have fallible sensory systems and they gen- species as models. Mimicry rings have not previously eralize from their predation experiences. Thus, although been modelled; we present an evolutionary simulation to they are much more likely to mistake a perfect mimic investigate the above questions. Complete convergence for its model, they can still mistake approximate resem- is not observed although Batesian mimicry is shown to blances for the real thing. This gives a greater level of be important factor in the origin of mimicry rings. protection to a mimic the more accurate its mimicry be- Bees and wasps both possess yellow and black striped comes. However, if the appearances of two prey species warning colourations, making an obvious display of their are sufficiently distinct, the predator will never mistake unpleasantness. But why do they display the same pat- one for the other. Therefore, before mimics can undergo tern? Imagine a situation where they both have differ- a process of pattern refinement, an initial resemblance in ent patterns. A predator, such as a bird, would have to the eyes of the predator is needed at first so that it may sample many of each species in order to learn to avoid occasionally confuse the two. An approximate resem- them both. However, when sharing the same pattern, blance can be arrived at due to random drift, or fairly the predator will see them both as the same type of large major mutations (Turner 1988). prey and so will eat fewer individuals before develop- Population sizes have a considerable effect on mimicry. ing an aversion to them. Thus, each individual has less When two unpalatable species converge as Mullerian¨ chance of being eaten as it is protected by numbers. It mimics, they are doing so in order to increase the pop- therefore makes sense that two unpalatable (bad-tasting) ulation size of individuals with their type of warning species would become co-mimics by converging upon the coloration. It follows, then, that unpalatable species same colour pattern. This is assuming that the predator with higher population sizes are better defended against must learn which prey types to avoid, rather than hav- predators than those with lower population sizes (posi- ing innate aversions — an assumption in line with past tive frequency dependence). Conversely, Batesian mim- mimicry research. This type of relationship, in which ics are better protected by lower population sizes. This both species benefit, is known as Mullerian¨ mimicry is because higher populations would dilute the model’s (Muller¨ 1879). protection so much as to make predators more willing The honest displays of unpalatability given by species to risk eating a prey of that colour pattern (negative such as bees and wasps are ready targets for free-riders. frequency dependence) (Pilecki & O’Donald 1971). A good example is the harmless hoverfly, which has black The coevolutionary dynamics involved in the two sep- and yellow stripes. It is easy to imagine the benefit arate cases of mimicry differ considerably (Turner 1987; gained by a tasty species in being confused for an un- 1995). Mullerian¨ mimics converge upon the same colour palatable one. So, gaining protection without evolving a pattern. That is, they ‘move’ together serving both as costly toxin or sting, the species evolves the same warn- co-mimics and models. Batesian mimicry is a different ing colour as an unpalatable species. The species that matter. A Batesian mimic’s colour pattern ‘moves’ to- is being mimicked is termed “the model”. This type ward that of the model’s in order to gain protection. The of mimicry was the first to be described in the litera- model then attempts to escape the parasitic mimic by ture and is known as Batesian mimicry (Bates 1862). evolving its pattern ‘away’ from the invading mimic, as 2 in Artificial Life VIII, Standish, Abbass, Bedau (eds) (MIT Press) 2002. pp 186–191 individuals that are slightly less parasitized (due to mi- 1. All of the Mullerian¨ mimics in a given ecosystem nor pattern differences) are more likely to survive. The should eventually converge into one large ring in order Batesian mimic is, however, expected to usually keep up to gain maximum protection; in the resulting coevolutionary arms race (Nur 1970). This is because the selection pressure on a tasty species 2. If the Mullerian¨ mimics do not converge into one large to gain protection is generally greater than the selec- ring, then the presence of Batesian mimics could entice tion pressure on a model to reduce the dilution of preda- them to do so, by adverging upon the rings; tor aversions. This type of mimetic arms race has been termed advergence (Brower, Brower, & Collins 1972). Although there are many mathematical and stochas- tic models of mimicry in the biological literature, this topic has not yet received any attention in the artifi- cial life literature, apart from the authors’ own previ- ous work (Franks & Noble 2002). Mimicry rings have hardly been explored at all in any literature, mainly due to the difficulty that a mathematical or stochastic model would have in modeling the large and complex ecosystem that would be required. This is where evolving artificial life models become the most practical and promising ap- proach. To that end, we present a simulation model that explores the evolution of mimicry rings under varying conditions. The Simulation Model

Figure 1: Mimicry ring example. Three separate species To model prey we needed to create populations of of butterfly that all share the same pattern in a ‘yel- ‘agents’ which each have an appearance and palatibil- low’ ring in Sirena. Note the subtle differences in the ity level. Multiple populations of prey species were used patterns. in each experiment. Different species of prey were each assigned a fixed palatibility level on a scale between zero and one (least to most palatable), where 0.5 is neutrally Mimicry rings are Mullerian¨ mimetic relationships be- palatable. Each individual had two genes with values tween two or more species, and are common among but- compositely representing their external appearance (e.g., terflies and bumblebees. It is called a “ring” because a colour pattern) or phenotype.1 Both genes were con- each species in the relationship has an effect on each of strained to values from 1–200. The Euclidean distance the others. Plowright (1980) showed that there are five of one phenotype from another represented their level different patterns of bumblebee in north-west Europe, of similarity. Palatable species have values greater than each constituting a mimicry ring of several species. A 0.5, and unpalatable species have values lower than 0.5. good example of a mimicry ring is the ‘tiger’ pattern A population of abstract predators was modelled with shared by different species of Heliconius butterflies in a Monte Carlo reinforcement learning system, as used in Sirena (Mallet & Gilbert 1995), alongside other ring pat- a stochastic model by Turner et. al. (1984). The preda- terns (see, e.g., Figure 1). It would be highly profitable tor’s experience of each phenotype was represented by a for a palatable species to invade these rings as a Batesian score of attack probability, which was initialized to am- mimic, as it would have less of a dilution affect on the bivalence at 0.5. After eating prey of a particular phe- palatibility associated with the colour pattern. notype, the predator would make a post-attack update The co-existence of mimicry rings in nature has of the relevant probability according to the palatibility prompted the question: why do the co-mimics (in the of the individual consumed. The predator would use its same geographical region) not all converge into one large experience of different prey appearances to help it decide ring (Mallet & Gilbert 1995)? This question is posed be- on whether or not to attack them at the next opportu- cause all of the mimics would mutually, and optimally, nity. Unlike the stochastic model, the predator general- benefit from evolving into one large ring. The authors ized on the basis of experience and thus would also, to have previously shown (Franks & Noble 2002) in a sim- a lesser extent, update its scores for the closest neigh- ple, one-dimensional, three-species case that Batesian bouring phenotypes accordingly. The formula used for mimics can ‘chase’ unpalatable species to within con- 1 vergence range of each other, provoking a Mullerian¨ re- Clearly a two-dimensional representation of possible phe- lationship. This suggests a potential for Batesian mimics notypes is a simplification. We have also looked at the evo- lution of mimicry rings in higher dimensional spaces; similar to influence the dynamics of mimicry rings. conclusions are reached but there are significant difficulties The above points suggest two working hypotheses: in visualizing and conveying the results. in Artificial Life VIII, Standish, Abbass, Bedau (eds)(MIT Press) 2002. pp 186–191 3 generalizing and updating the probabilities after eating regardless of its decision. Random asexual reproduction a prey was: then took place amongst the surviving prey. Every gen- eration the oldest predator would die, and be replaced P1 = (P0 + α(λ − P0)) × W (1) with a new predator with its memory initialized to the naive attack level of 0.5 representing ambivalence. All of This produces an updated palatability score P1, based the experiments were run over 200,000 generations and on the previous score P0, for phenotypes at a given dis- prey populations were kept constant after reproduction. tance from the consumed prey whose palatability was λ. Mutation was implemented as follows: α is a variable learning rate and is calculated using: 1. A random direction in the multidimensional space is α = 0.5 + |λ − 0.5| (2) chosen by selecting a random number from a normal 2 W is a weighting (for generalization) which is calcu- distribution (0 mean, unit variance) for each gene. lated according to the distance of the phenotype from the consumed prey’s phenotype, and is calculated with: 2. The distance is then selected over a normal distribu- tion (0 mean, unit variance) to allow for varying mu- G − D tation sizes, with a bias towards smaller ones. W = (3) G 3. The offspring then mutates in the selected direction at Where G is the predator’s generalization rate and D is the selected distance. the Euclidean distance between the phenotype and the consumed prey’s phenotype. Note that the operator is performed on every offspring. The above equations have the effect of updating a The result is that most offspring are slight variants or predator’s opinion of a particular phenotype. The rate replicas of their parent. This method was used in or- of learning is dependent on how far the predator’s cur- der to avoid the orthogonal bias present with stepwise rent opinion is from the palatibility of the consumed mutation operators. Also, the issue of mutation bias prey (equations 1 and 2). The predator’s memory is due to unnatural boundaries (Bullock 1999) was han- updated by generalizing over similar phenotypes within dled using a wrap-around function (making alleles 1 and the predator’s generalization range from the consumed 200 neighbours) on prey mutations and predator gener- prey’s phenotype (equation 3). This update can be imag- alization. These mutation issues are important to any ined as a cone shape, where the tip of the cone is the simulation model as they can help to avoid artefactual update for the consumed phenotype. The further away results. They are, however, of particular significance to from the consumed phenotype the pattern is, the less it mimicry models as random-drift and mutation are vital is updated. to the initiation of mimicry. Before being used to express the probability of con- suming a prey item, scores were transformed with the Results logistic function, which meant that predators were more For our purposes, a mimicry ring is defined as a mimetic decisive about prey for which they had a relatively strong relationship between two or more unpalatable species. opinion. Predators made decisions about whether or not A mimetic relationship is defined as a Euclidean dis- to eat a prey by comparing the transformed score to a tance of less than fifteen between the modes of the two random number in the range of zero to one. This meant species. The distance of fifteen was calculated by multi- that a predator could become averse to eating some or plying the generalization rate by two and deducting one even all of the prey species — note that we must assume (which gives the minimum distance of overlapping gener- that an alternative food source is available as predators alization), and then multiplying that by 1.5 (to allow for in the model will not starve even if they refuse all prey. spread of individuals within a species and generalization To implement predator memory degradation over time, over each of them), which gave clusters that matched each predator’s memory gradually reverted back toward that of manual observations for the experiments. ambivalence (0.5) at a constant rate of two percent per Experiment 1 was conducted with 20 unpalatable prey prey offering. species. Each had a palatibility of 0.1 and population The predator population size was kept constant size of 300. After 200,000 generations of evolution, the throughout the experiments at 160. Each predator was number of co-existing mimicry rings formed were tallied. presented with 10 potential prey items per prey gener- Figure 2 shows that five or six mimicry rings usually ation. Prey individuals were randomly selected in pro- evolve, and no run resulted in less than four rings. Figure portion to their relative population numbers from across 3 shows the position of the initial modal phenotypes as all prey populations. The predator would then make well as the final coevolved modal phenotypes. a probabilistic choice of whether or not to eat the prey based on its experience of the phenotype. After being of- 2If a uniform distribution was used here, it would give a fered a prey item the predator’s memory would degrade, bias to diagonal directions. 4 in Artificial Life VIII, Standish, Abbass, Bedau (eds) (MIT Press) 2002. pp 186–191

12 200 11 10 9 150 8 7 6 100

5 Gene 2

Frequency 4 50 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 Number of rings in ecosystem Gene 1

Figure 2: Frequency of mimicry rings after 200,000 gen- Figure 3: Initial random and final evolved position of erations, with 20 unpalatable species with equal popula- each prey species’ modal phenotype for a typical run tions of 300 and palatibility levels of 0.1. Results were with equal population numbers. Circles represent start- recorded over 20 runs. ing phenotypes and squares represent end phenotypes. Note the formation of six mimicry rings and one solitary species. Experiment 2 was conducted with the same condi- tions as experiment 1 except that four palatable species, of population size 300 and palatibility 0.9, were also included to allow for Batesian mimetic relationships. with evolutionary simulation models, as they do not rely Figure 4 shows strong shift toward a lower number of finding an equilibrium. A further examination of the re- mimicry rings, with four rings being the most common. sults showed that in all runs there is very little (if any) Figure 5 shows the position of the initial modal pheno- change from generation 150,000 to 200,000, suggesting types as well as the final modal phenotypes. Different that the ecosystem has reached a steady state and the population sizes were tried for the two experiments, and various rings are not about to collapse into one large ring gave similar results. given more time. Such an explanation for the diversity A statistical comparison of the number of mimicry of mimicry rings in nature has been suggested in previ- rings found in each of the two experiments indicated a ous mimicry literature (see e.g., Turner , 1977; Sheppard significant difference (t = 7.333, p < 0.001) with more et. al., 1985 ). However, observations of the individual rings being found in experiment 1 than in experiment 2. simulation runs shows that the coexisting rings are not always far apart. The diversity could perhaps also be Discussion explained by the fact that a mutation is unlikely to take Experiment 1 shows that, under conditions in which all an individual from being a perfect mimic of its species’ species are (equally) unpalatable, hypothesis 1 was not current ring to being a perfect mimic of another ring; borne out and thus, at least with the population sizes being an imperfect mimic of another ring is not as se- used, a single large mimicry ring should perhaps not be lectively advantageous as being a perfect mimic of the expected in nature. This shows that current mimicry the- original ring. Most of the time a mutation will result ory explains the co-existence of multiple mimicry rings in an individual no longer being protected by numbers. without the need to bring in other selective forces such The simulation shows that current mimicry theory alone as sexual selection. Results show that mimics cluster in does explain the coexistence of multiple mimicry rings. rings in different areas of phenotypic space, sometimes The results of experiment 2 show that the introduction far enough apart so that it would be rare for a large of palatable species induced a smaller number of mimicry enough mutation to bridge the gap. Thus, the pheno- rings, supporting hypothesis 2. This is because, due to types of the rings are sufficiently different so that the pressure to evolve a different phenotype from the modal predator would not mistake a member of one for a mem- phenotype of the species (because of negative frequency ber of another. Of course, on freak occasions it would dependence), the palatable species would inevitably be- be possible for an individual to mutate across the gap. come Batesian mimics. The presence of Batesian mim- However, it is very unlikely and not guaranteed to be ics would provide positive selection pressure on mutants in the right ‘direction’ or beneficial to the individual. and would, therefore, increase the probability that they Such a state would be well suited to further examination would evolve an initial resemblance to another unpalat- in Artificial Life VIII, Standish, Abbass, Bedau (eds)(MIT Press) 2002. pp 186–191 5

12 200 11 10 9 150 8 7 6 100

5 Gene 2

Frequency 4 50 3 2 1 0 0 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 Number of rings in ecosystem Gene 1

Figure 4: Frequency of mimicry rings after 200,000 gen- Figure 5: Initial random and final evolved position of erations, with 20 unpalatable species and four palatable each prey’s modal phenotype for a typical run with four species of palatibility 0.9 and population size 300 added. palatable species added. Circles represent starting phe- Results were recorded over 20 runs. Notice how there notypes and squares represent final phenotypes. Palat- are fewer rings when palatable species are present. able species are not shown; unpalatable species form two large rings and three small rings. Notice how there are fewer rings when palatable species are present. able species. Also, Batesian pressure on mimicry rings has the potential to push one ring into the range of an- other, helping to bridge a large phenotypic difference between them. Once mimicry rings have enough mem- actions of the Linnean Society London 23:495–566. bers they can ‘tolerate’ the presence of a Batesian mimic. Brower, L. P.; Brower, J. V. Z.; and Collins, C. T. It appears that situation will persist, unless additional 1972. Parallelism, convergence, divergence, and the Batesian mimics invade in sufficient numbers to desta- new concept of advergence in the evolution of mimicry. bilize the ring. As a result, Batesian mimics can induce In Deevey, E., ed., Ecological essays in honour of G. the convergence of nearby mimicry rings. Evelyn Hutchinson. Connecticut Academy of Arts and Although prey in the model presented had only two Science. 57–67. genes, work in progress shows that the same conclusions Bullock, S. 1999. Are artificial mutation biases unnat- seem to hold for a four-gene case. However, as noted ear- ural? In Floreano, D.; Nicoud, J.-D.; and Mondada, lier, the two-gene case is easier for display purposes. Fu- F., eds., Fifth European Conference on Artificial Life ture work will endeavour to co-evolve the population of (ECAL99), 64–73. Springer, Heidelberg. predators with the prey, and to move away from the stan- Franks, D. W., and Noble, J. 2002. Conditions for the dard assumption in the mimicry literature that predators evolution of mimicry. In Proceedings of the Seventh learn from a blank slate. This should help to settle any International Conference on Simulation of Adaptive misgivings about the arbitrariness of predator attributes Behavior. MIT Press, Cambridge, MA. such as generalization distance, learning rate and for- Mallet, J., and Gilbert, L. E. 1995. Why are there so getting rate. Co-evolving predators with prey may also many mimicry rings? Correlations between habitat, influence the emergence of mimicry rings. behaviour and mimicry in Heliconius butterflies. Bio- logical Journal of the Linnean Society 55:159–180. Acknowledgments Muller,¨ F. 1879. Ituna and Thyridia; a remarkable case of mimicry in butterflies. Trans. Entomol. Soc. Lond. Thanks to John Turner for his expert advice on the- May 1879:xx–xxix. ories of mimicry; any misrepresentation of his ideas is Nur, U. 1970. Evolutionary rates of models and mimics our responsibility. Thanks also to Seth Bullock for help in Batesian mimicry. American Naturalist 104:477– with the mutation operator and to Dave Harris for useful 486. comments on the draft. Pilecki, C., and O’Donald, P. 1971. The effects of preda- tion on artificial mimetic polymorphisms with perfect References and imperfect mimics at varying frequencies. Evolu- Bates, H. W. 1862. Contributions to an insect fauna of tion 55:365–370. the Amazon valley. Lepidoptera: Heliconidae. Trans- Plowright, R. C., and Owen, R. E. 1980. The evolu- 6 in Artificial Life VIII, Standish, Abbass, Bedau (eds) (MIT Press) 2002. pp 186–191

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